Question $2-$ Consider a case where a binary system of components is in vapor$-$liquid equilibriums and the vapor phase is consider to be an ideal gas and the liquid is a non$-$ideal mixture. We consider the two components to be methanol $($component $1)$ and water $($component $2)$; for this system, we can use the Wilson equation to define the activity coefficients: USE MATLAB CODING

$ln{}_1=-ln(x_1+{}_{12}{x}_2)+x_2(\frac{{}_{12}}{x_1+{}_{12}{x}_2}-\frac{{}_{21}}{{}_{21}{x}_1+x_2})$

$ln{}_2=-ln(x_2+{}_{21}{x}_1)-x_1(\frac{{}_{12}}{x_1+{}_{12}{x}_2}-\frac{{}_{21}}{{}_{21}{x}_1+x_2})$

Where,

${}_{12}=\frac{v_2^L}{v_1^L}$exp$(-\frac{A_{12}}{R}T)$

${}_{21}=\frac{v_1^L}{v_2^L}$exp$(-\frac{A_{21}}{R}T)$

The mole fraction definitions and phase relations give us the following $4$ equations:

$x_1+x_2=1,y_1+y_2=1$

$P{y}_1={10}^{(5\mathrm{.}20409-\frac{1581\mathrm{.}341}{T-3\mathrm{.}50})}{}_1{x}_1,P{y}_2={10}^{(4\mathrm{.}6543-\frac{1\mathrm{.}425\mathrm{.}264}{T-64\mathrm{.}848})}{}_2{x}_2$

Using MATLAB to determine $x_1,x_2,y_1,$ and $y_2$ given that $T=363,R=1\mathrm{.}987,P=1,A_{12}=19\mathrm{.}2547,$

$A_{21}=554\mathrm{.}0494,v_1^L=40\mathrm{.}73,v_2^L=18\mathrm{.}07$